dV k Ap
dt L
(3.4-7 )
This equation relating the flow rate of a filtrate with viscosity μ through a bed of thickness L and face area A to the driving pressure Δp is called Darcy’s basic filtration equation.
equation (3.4-11 ) for Rm in equation (3.4-9 ) gives
dV
Ap
dt dV Ap
dt r(LLm)
r(L Lm )
3.4-12
Equation (3.4-12) relates the flow rate of filtrate to the pressure drop,
)
➢ When the pressure difference applied to the cake increases
The rate of filtration will (
)
➢ When the temperature of the suspension increases
The rate of filtration will (
Most cakes, however, are compressible and their specific resistance changes with the pressure drop across the cake.
r r0p n
3.4-13
5. Volume of the cake deposited per unit volume of filtrate obtained
problem
• Fluid passes through a packed bed in laminar flow, the basic equation applied to packed bed for relating the pressure drop with the width of bed, average interstitial velocity, diameter of tortuous channels, and the properties of fluids is
3.4.2 Flow Rate-Pressure Drop Relationships
At the beginning of batch cake filtration, the whole pressure drop available is across the medium itself since as yet no cake is formed.
3. Liquid viscosity
The liquid viscosity μ is constant provided that the temperature remains constant during the filtration cycle and that the liquid is Newtonian.
As the pores in the medium are normally small and rate of flow of filtrate is low, laminar flow conditions are almost invaonship between the superficial velocity u and the pressure drop can be described by Kozeny-Carman equation
4. Specific cake resistance
The specific cake resistance r should be constant for incompressible cakes but it may change with time in the case of variable rate filtration, because of variable approach velocity.
)
equation
• Basic filtration equation comes from
(
) equation, and constant
k depends on the (
) of
packed bed and (
) of
particle, and specific cake resistance r is
)
➢ When the concentration of the suspension increases
The rate of filtration will (
)
p
150uL Dp2 2
1
3
2
3.4-2
The filtration rate is
dV
3
Ap
dt
4.17
sp vp
2
1
2
L
3.4-5
Let
3
k
4.17
sp vp
2
1
2
3.4-6
Where k is a constant referred to as the permeability of the bed and substitutes equation (3.4-6 ) into equation (3.4-5 )
The volume of the particles deposited per unit volume of filtrate is defined by c.
cV A
c Vc V
Where c is constant, which depends on the concentration of solids in the slurry and the porosity of cake.
As the resistance of the cake may be assumed to be directly proportional to the amount of cake deposited (only true for incompressible cakes) it follows that for a given filtration area A.
• Hagen-Poiseuille modified by substituting
equivalent diameter of channel for the
diameter, and superficial velocity for the
interstitial velocity gives a(
Filter medium resistance
In the cake filtration, two resistances are presented in series, one of which, the cake resistance R increases and other, the medium resistance Rm may be assumed constant with time. Equation (3.4-8 ) becomes:
R=rL
3.4-10
Where L is the width of cake deposited and r is the specific cake resistance.
Similarly for the filtration medium of width Lm
Rm=rLm
3.4-11
Substitution of equation (3.4-10 ) for R and
c can be related to the cumulative volume of filtrate V and the width of cake L
L cV A
3.4-14
6. Medium resistance
The medium resistance Rm should normally be constant but it may be vary with time as a result of some penetration of solids into the medium and sometimes it may also change with applied pressure because of the compression of fibres in the medium.
dV
Ap
3.4-9
dt (R Rm )
In practice, however, the assumption made above that the medium resistance is constant is rarely true because some penetration and blocking of the medium inevitably occurs when particles impinge on the medium
(
)k
For an incompressible cake
➢ When the filtrate increases
The rate of filtration will (
)
➢ When the filtration area increases
The rate of filtration will (
The thickness of cake deposited and other parameters, some of which can , in certain circumstances, be assumed to be constant.